Detection of One-Lung Intubation Incidents

Abstract

Lung sounds are very common source for monitoring and diagnosis of pulmonary function. This approach can be used for detecting one lung intubation (OLI) during anesthesia or intensive care. In this paper, an algorithm for detecting OLI from lung sounds is presented. The algorithm assumes a multiple-input-multiple-output system, in which a multi-dimensional auto-regressive model relates the input (lungs) and the output (recorded sounds). An OLI detector is developed based on the generalized likelihood ratio test (GLRT), assuming coherent distributed sources for each lung. This method exhibited reliable results also when the lungs were modeled by incoherent distributed sources, which is a more accurate model for lung sources. The algorithm was tested using real breathing sounds recorded in an operating room, and it achieved an OLI detection rate of more than 95%, for each breathing cycle.

Appendix A: Proof of (9)

Maximization of (8) with respect to the unknown matrix, A, is achieved via equating the corresponding partial derivative to zero. The last term of (8) is relevant to calculate the derivative of the log-likelihood with respect to A:

$$ \begin{aligned} &-\frac{1}{2}\sum_{n=1}^N {\left[ {\left( {{\mathbf{y}}[n]-{\mathbf{Ay}}^{\left( M \right)}[n]} \right)^{T}{\mathbf{R}}^{-1}\left( {{\mathbf{y}}[n]-{\mathbf{Ay}}^{\left( M \right)}[n]} \right)}\right]}\\ =&-\frac{1}{2}\sum_{n=1}^N \left[{{\mathbf{y}}^{T}[n]{\mathbf{R}}^{-1}{\mathbf{y}}[n]-{\mathbf{y}}^{\left( M\right)T}[n]{\mathbf{A}}^{T}{\mathbf{R}}^{-1} {\mathbf{y}}[n]-{\mathbf{y}}^{T}[n]{\mathbf{R}}^{-1}{\mathbf{Ay}}^{\left( M \right)}[n]+\left( {{\mathbf{Ay}}^{\left( M \right)}[n]} \right)^{T}{\mathbf{R}}^{-1}{\mathbf{Ay}}^{\left( M \right)}[n]}\right]. \\ \end{aligned} $$

(28)

The derivative of a scalar c with respect to a matrix A is defined as a matrix whose ij-th element is given by $\left[ {\frac{\partial c}{\partial {\bf A}}} \right]_{ij} =\frac{\partial c}{\partial A_{ij} }.$ We shall use the following identities (for proof of non-trivial identities, see Makhoul11):

$$ \frac{\partial }{\partial {\mathbf{A}}}\left[ {{\mathbf{x}}^{T}{\mathbf{Ay}}} \right]=\frac{\partial }{\partial {\mathbf{A}}}\left[ {{\mathbf{y}}^{T}{\mathbf{A}}^{T}{\mathbf{x}}} \right]={\mathbf{xy}}^{T}, $$

(29)

$$ \frac{\partial }{\partial {\mathbf{A}}}\left[ {({\mathbf{Ax}})^{T}{\mathbf{C}}({\mathbf{Ax}})} \right]=({\mathbf{C}}+{\mathbf{C}}^{T})({\mathbf{Ax}}){\mathbf{x}}^{T}. $$

(30)

The derivative of (28) w.r.t. A is calculated using (29) and (30), and thus, the derivative of (8) can be written as

$$ \begin{aligned}\frac{\partial }{\partial {\mathbf{A}}}\hbox{log}(f({\mathbf{y}}[1],\ldots,{\mathbf{y}}[N]|{\mathbf{R}},{\mathbf{A}})=\\ -\frac{1}{2}\sum_{n=1}^N {\left[ {-{\mathbf{R}}^{-1}{\mathbf{y}}[n]{\mathbf{y}}^{\left( M \right)T}[n]-{\mathbf{R}}^{-T}{\mathbf{y}}[n]{\mathbf{y}}^{\left( M \right)T}[n]+\left( {{\mathbf{R}}^{-1}+{\mathbf{R}}^{-T}} \right){\mathbf{Ay}}^{\left( M \right)}[n]{\mathbf{y}}^{\left( M \right)T}[n]} \right]}.\\ \end{aligned} $$

(31)

In order to find the ML estimator of A, the above derivative should be equated to zero. Since R is a covariance matrix, then ${\mathbf{R}}={\mathbf{R}}^{T}$ and ${\mathbf{R}}^{-1}={\mathbf{R}}^{-T}.$ Therefore we obtain the following equation:

$$ -\frac{1}{2}\sum_{n=1}^N {\left[ {-2{\mathbf{R}}^{-1}{\mathbf{y}}[n]{\mathbf{y}}^{\left( M \right)T}[n]+2{\mathbf{R}}^{-1}\hat{{\mathbf{A}}}_{\rm ML} {\mathbf{y}}^{\left( M \right)}[n]{\mathbf{y}}^{\left( M \right)T}[n]} \right]} ={\mathbf{0}}. $$

(32)

Equation (9) can be obtained by solving (32) for $\hat{{\mathbf{A}}}_{\rm ML}$ under the assumption that the matrix $_{ }\sum_{n=1}^N {{\mathbf{y}}^{\left( M \right)}[n]{\mathbf{y}}^{\left( M \right)T}[n]}$ is invertible.$\hfill\square$